OPTIMIZATION OF OPENING SIZE FOR CASTELLATED BEAM …ijoce.iust.ac.ir/files/site1/user_files_5jkw45/eng/tahabakhshpoori... · OPTIMIZATION OF OPENING SIZE FOR CASTELLATED BEAM

INTERNATIONAL JOURNAL OF OPTIMIZATION IN CIVIL ENGINEERING

Int. J. Optim. Civil Eng.,2015; 5(3):301-313

OPTIMIZATION OF OPENING SIZE FOR CASTELLATED BEAM

WITH SINUSOIDAL OPENINGS

P.D. Kumbhar*, † and A.M. Jamadar Rajarambapu Institute of Technology, Islampur, Maharashtra, India

ABSTRACT

Castellated beams are generally provided with hexagonal and circular openings in the web

portion. However, in view of structural applications, appropriate size and shape of openings

in web are always a major issue of concern. Research work carried out in optimizing sizes of

castellated beam with hexagonal openings have reported that castellated beams fail mainly

by local failure modes and stress concentrations at opening edges. Castellated beams with

sinusoidal openings offer better performance due to its increased area for stress distribution

in addition to curved edges that causes smooth stress distribution. Few researchers have

studied flexural behaviour of castellated beams with sinusoidal openings; however,

optimization for size of such openings has not been reported so far.

The paper focuses on parametric study of castellated beam with sinusoidal openings for

optimization of opening size. Finite element analysis (FEA) is carried out by Abaqus

software and also by Eurocode for different opening sizes and results obtained is

experimentally validated. Results show that, castellated beam with sinusoidal opening of

size 0.55times the overall depth of beam gives better strength.

Keywords: castellated beam; cellular beam; finite element analysis; Abaqus; web openings;

sinusoidal openings; optimization of web openings.

Received: 10 March 2015; Accepted: 21 June 2015

1. INTRODUCTION

Use of steel for structural purpose in structure is rapidly gaining interest these days. In steel

structures the concept of pre-engineered building (PEB) is most popular due to its ease and

simplicity in the construction. Pre-engineered buildings have very large spans but

comparatively less loading. Generally, steel sections satisfy strength requirement but the

*Corresponding author: Rajarambapu Institute of Technology, Islampur, Maharashtra, India

†E-mail address: [email protected] (P.D. Kumbhar)

P.D. Kumbhar and A.M. Jamadar

302

difficulty is that, section have to satisfy serviceability requirement (i.e. deflection criteria in

safety check). This necessitates the use of beams with greater depth to satisfy this

requirement. Use of castellated beams is the best solution to overcome this difficulty [15].

The castellated or perforated web beam is the beam which has perforation or openings in its

web portion. Generally, the castellated beams are with hexagonal (Fig. 1) or square or

circular shaped openings. The beams with circular openings are called as cellular beams

(Fig. 2) [1]. As a modification, in castellated beam with hexagonal opening, the corners of

opening are made round so as to offer smooth stress transfer area to avoid stress

concentration. The beams with such curved shaped openings are known as castellated beam

with sinusoidal openings (Fig. 3) [4].

Use of castellated beam with hexagonal openings is very common in recent years because of

the simplicity in its fabrication. Castellated beams are fabricated by cutting flange of a hot

rolled steel I beam along its centerline and then welding two halves so that the overall beam

depth gets increased for more efficient structural performance against bending [5].

Castellated beams with hexagonal openings and circular openings have found widespread

use, primarily in buildings, because of great savings in materials and construction costs. The

research studies report that, use of beams with hexagonal openings require smaller amount

of steel material and it is also superior to cellular beams from the cost point of view [6].

1.1 Types of castellated beam

Castellated beams are generally classified on the basis of type or shape of perforations made

in the web of beam. Based on the shape of the opening the various types of castellated

beams are shown in Fig. 1 to Fig. 3 [1].

Figure 1. Hexagonal Castellated Beam Figure 2. Cellular Beam

Figure 3. Castellated Beam with Sinusoidal Shaped Opening

1.2 Terminology in castellated beam

The various basic terms involved in the analysis and design of castellated beams are

illustrated in Fig. 4, adopted from [9].

OPTIMIZATION OF OPENING SIZE FOR CASTELLATED BEAM WITH …

303

Figure 4. Typical cross section of the beam

Various dimensional parameters involved in the castellated beam with sinusoidal openings

(Fig. 4) are defined as given below:

Do = Depth of opening provided.

D = Overall depth of the opening.

S = C/C spacing between the two opening

e = Clear distance between two opening

b = Width of flange of I beam

tf = Thickness of flange of I beam

tw = Thickness of web of I beam

2. REVIEW OF PREVIOUS STUDIES

In recent times, a lot of research work has been carried out for analysis and design of

castellated beams, especially with hexagonal openings. There is no universally accepted

design method for castellated beam because of complexity in geometry accompanied by

complex mode of failure [11]. At present, there are possibly six failure modes of castellated

beam namely, formation of flexure mechanism, lateral torsional buckling, formation of

Vierendeel mechanism, rupture of welded joint, shear buckling of web post and compression

buckling of web post[14].Various research studies carried out for analysis and design of

castellated beams are presented in the following section.

2.1 Wakchaure and Sagade [14]

Researchers have studied the flexural behavior of steel I –shaped castellated beams with

hexagonal openings. The beams were modeled for different depths of web openings by

varying spacing (S) to opening depth (Do) ratio (i.e. S/Do). The analysis of castellated steel

beams (I-shaped cross-section), is performed using finite element software package

ANSYS14 with two point load and simply supported support condition. The deflection at

center of beam and study of various failure patterns are studied. The beams with increase in

depth are then compared with each other and with parent section for various parameters and

for serviceability criteria. From the results, it is concluded that, the castellated steel beam

behaves satisfactorily with regards to serviceability requirements up to a maximum web

opening depth of 0.6 times the overall depth (D). Castellated beams have proved to be

efficient for moderately loaded longer spans where the design is controlled by deflection.


304

2.2 Wakchaure, Sagade and Auti [13]

The experimental investigations have been carried out by the researchers to study the

behavior of castellated beams under two point loading (four point bending) by varying depth

of hexagonal openings (and hence the overall depth). The results indicate that beam with

opening size of 0.6 times the overall depth (D) carries maximum load compared to other

sizes of openings. Also, investigators have concluded that with increase in depth of opening,

vierendeel failure of beam becomes predominant.

2.3 Soltani, Bouchaïr and Mimoune [11]

The researchers have prepared a nonlinear numerical model to obtain behavior of castellated

beams with hexagonal and octagonal openings. Parametric study is also carried out by

varying depth of openings with increments of 10mm. The numerical results are compared

with the existing literature and validated with help of MSC/NASTRAN software. The failure

patterns of beams with various sizes are also studied. It is concluded that, the castellated

beam with octagonal openings are more susceptible to local failure of web post buckling

than the castellated beam with hexagonal opening.

2.4 Erdal and Saka [5]

The researchers have studied the load carrying capacity of optimally designed castellated

beam by varying number of holes and spacing. FEA of same beams is also carried out by

applying central point load and failure patterns are studied and verified using ANSYS. Study

shows that, even though the members are relatively of shorter spans, lateral supports are

governing factor for the analysis of beams due to torsional buckling. It is concluded that, the

beam fails in vierendeel mode when the load is applied above openings, while it fails in the

web post buckling when load is applied in between space of the openings.

3. ANALYSIS AND DESIGN OF CASTELLATED BEAM AS PER

EUROCODE 3

The materials presented in this section are based on Refs. [2, 3, and 8].

3.1 Guidelines for perforations in web

The perforations made in the web greatly affecting the structural performance of the beam.

Therefore, it is essential to make some logical and practical considerations while providing

perforations in the beam. Following are the general guidelines which are given by Eurocode

and some of them are based on the field or practical considerations. These standards in web

perforations can be changed or modified without affecting the structural performance of the

beam. These guidelines are as follows;


305

3.2 Design of castellated beam

In this section, design standards provided by Eurocode 3 for designing of castellated beam

are illustrated.

3.2.1 Check for moment (flexural) capacity of the beam

In this check, we have to ensure that maximum moment induced in the beam due to external

loads should be less than moment capacity of the upper and lower Tee.

(1)

(2)

Where,

Mu = Maximum moment induced in the beam as per loading conditions.

MpTee = Moment capacity of the upper or lower Tee.

ATee = Area of upper or lower Tee.

Py = Yield stress of steel.

z = Lever arm (Distance between the centroid of upperand lower Tee).

3.2.2 Check for shear capacity of the beam

Maximum vertical and horizontal shear induced in the beam due to external loading should

be less than vertical and horizontal shear capacities of the beam respectively.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Where,

Vmax = Maximum vertical shear.

Hmax = Maximum horizontal shear.

Pv = Shear strength of castellated beam.


306

Av = Shear area (shear area of whole cross section)

= (D-2tf) × tw (9)

Pvy = Vertical shear capacity.

Awt = Shear are of Tee

= (D - 2tf-Do) × tw (10)

Pvh = Horizontal shear capacity.

Amwt = Horizontal shear area

= e × tw (11)

VH = Horizontal shear.

T = Axial load at different point.

M = Bending moment at different point.

3.2.3 Check for flexural and buckling strength of web post

Flexural and buckling strength of web post should be checked by the empirical formula

given below,

[ (

) (

)

] (12)

Where,

Mmax = Bending moment of critical web post section

= (

) (13)

Me = Bending resistance of critical web post section

= ( )

(14)

C1 = (

) (

)

(15)

C2 = (

) (

)

(16)

C3 = (

) (

)

(17)

3.2.4 Check for Vierendeel bending of tee

Vierendeel bending moment of the lower or upper Tee should be less than the local bending

resistance of respective Tee.


307

(18)

(19)

MpTeeLocal = Bending resistance of Tee of beam.

Mpv = Vierendeel bending moment.

leff = Effective length of opening.

Effective length of opening is depends on the type of opening provided.

For Circular opening, leff=0.45×Do (20)

For other opening leff is width of opening.

3.2.5 Check for fracture in welding

Strength of weld should be more than maximum horizontal shear force in the section,

(21)

(22)

3.2.6 Check for deflection

Deflection of beam is calculated as per standard formulae for perforated depth of the beam.

Additional deflection due to openings is calculated by adding 15% to 25% deflection in

above calculated deflection.

4. METHODOLOGY ADOPTED FOR OPTIMIZATION OF OPENINGS

Following the design standards of Eurocode, the approach is decided to achieve objectives

of the research. The analysis of the beam with sinusoidal shaped openings is carried out for

different sizes and the optimized section is tested experimentally for the purpose of

validation of the research.

4.1 Selection of method of analysis

In order to optimize the dimension of the openings of the castellated beam, it is very

important to decide proper analytical method. Due to complex geometry of castellated beam

the finite element analysis (FEA) is the best available method to analyse the beams. FEA of

castellated beamunder consideration is done by simulation software “Abaqus/CAE 6.13”.

4.2 Selection of parameters for parametric study on beam with sinusoidal openings

Depending upon the limitations of opening specified by the codes, different dimension of

the sinusoidal openings are selected. The parameter considered for the study is D/Do ratios

and S/Do of the opening. The variations in the parameters and corresponding cross sectional


308

dimensions of the sinusoidal openings are given in Table -1. All the castellated beams have

been derived from the 100 mm depth hot rolled steel (HRS) I section. The analysis of all the

beams considering the parameters given in Table -1 are modelled and analysed in Abaqus

software and the optimized section is found out. While varying the S/Do ratio for sinusoidal

openings beam, it is observed that, as S/Do ratio reduces (from 1.4 to 1.2) the clear distance

between two openings (i.e. „e‟) goes on reducing. So beam is tended to fail in local failure of

horizontal shear. For sinusoidal beams, first of all S/Do (i.e. 1.4) ratio is kept same while

D/Do ratio is kept varying from 1.25 to 1.75. After this same process is repeated for all other

S/Do ratio (i.e. for S/Do = 1.3 and 1.2).

Table 1: Parameters considered for sinusoidal shaped opening

Sr. No. D (mm) Do(mm) D/Do S/Do S (mm) e (mm)

1 140 80 1.75 1.4 112 22.65

2 145 90 1.61 1.4 126 22.09

3 150 100 1.5 1.4 140 26.36

4 155 110 1.41 1.4 154 32.12

5 160 120 1.33 1.4 168 38.05

6 140 80 1.75 1.3 104 14.34

7 145 90 1.61 1.3 117 14.66

8 150 100 1.5 1.3 130 16.69

9 155 110 1.41 1.3 143 17.08

10 160 120 1.33 1.3 156 23.2

11 140 80 1.75 1.2 96 7.06

12 145 90 1.61 1.2 108 7.04

13 150 100 1.5 1.2 120 9.95

14 155 110 1.41 1.2 132 10.1

15 160 120 1.33 1.2 144 11.94

4.3 Failure criteria

In order to decide the failure of the beam in Abaqus, Von-misses failure criteria has been

used. The criteria states that, failure of the structure would take place if the von-misses

stresses in the structure reaches to the value of yield stress of the material. Thus, in the

present work, as steel is the material used for beams and yield stress of steel is 250 N/mm2

and hence the load corresponds to yield stress of steel is taken as permissible load.

5. FINITE ELEMENT ANALYSIS OF CASTELLATED BEAM WITH

SINUSOIDAL OPENINGS

FEA of all castellated beams is carried out in Abaqus software to determine the optimum

section which fails at greater load.


309

Figure 6. FE model of castellated beam Figure 7. Meshing of castellated beam

FE model of one of such castellated beams with sinusoidal openings is shown in Fig. 6

along with loading and boundary condition. The beam is modelled as 3D shell element and

the meshing of model is shown in Fig. 7. Quad-dominated S4R doubly curved element is

used for meshing purpose. The various dimensions of openings along with their loads at

yielding, deflections by FEA and their respective stresses for yield load are given in Table

2.for sinusoidal shaped openings [7].

Table 2: Results of FEA of castellated beam with sinusoidal openings

Sr. No. Do D D/Do S/Do Length of

Opening

Load at

Yield(kN)

Deflection by

Software (mm)

Stresses

(N/mm2)

1 80 140 1.75 1.4 89.35 31 1.468 252.913

2 90 145 1.61 1.4 103.91 32 1.309 257.845

3 100 150 1.5 1.4 113.64 32 1.515 251.9

4 110 155 1.41 1.4 121.88 34 1.594 251.03

5 120 160 1.33 1.4 129.95 30.6 1.914 259.026

6 80 140 1.75 1.3 89.66 29.3 1.287 257.242

7 90 145 1.61 1.3 102.34 29.6 1.291 255.771

8 100 150 1.5 1.3 113.31 30.6 1.402 253.179

9 110 155 1.41 1.3 125.92 31 1.353 251.461

10 120 160 1.33 1.3 132.8 29.7 1.59 259.808

11 80 140 1.75 1.2 88.94 14.1 0.584 250.564

12 90 145 1.61 1.2 100.96 14.9 0.614 253.944

13 100 150 1.5 1.2 110.05 21 0.85 251.262

14 110 155 1.41 1.2 121.9 25 1.155 252.25

15 120 160 1.33 1.2 132.06 23 1.03 254.123

From the result of this analysis it is observed that the beam with depth of opening 0.55

times its overall depth behaves satisfactorily in respect of load carrying capacity (32.2kN).

In the other words beam with D/Do ratio of 1.41 and S/Do ratio of 1.4 gives more satisfying

results than the other. The variation in failure load against the depth of opening is illustrated

in graphical format in Fig. 8. While Fig. 9 and Fig. 10 show the variation in stress

(maximum 251.03 N/mm2) and deflection (maximum 1.594 mm) respectively for optimized

castellated beam with sinusoidal shaped opening. This optimized beam is highlighted in

above Table 2.


310

Figure 8. Variation in yield load for different S/Do and D/Do ratio for sinusoidal opening

Figure 9. Variation in stresses of optimized castellated beam with sinusoidal openings

Figure 10. Deflection of optimized castellated beam with sinusoidal openings

6. EXPERIMENTAL VALIDATION OF OPTIMIZED BEAM

Optimized beam which are obtained by finite element analysis are fabricated and tested in

the laboratory under the two point loading applied by universal testing machine (UTM) of

capacity 600 kN. The test setup for both the optimized castellated beam with sinusoidal

opening is shown in Fig. 11. Deflection is measured at centre by using dial gauge. The

results of these tests are given in Fig. 12 graphical form by load vs. deflection curve.

10

20

30

40

75 95 115 135

Lo

ad

(k

N)

Depth of Opening (mm)

Variation in Load Against Depth Of Opening

S/Do=1.4S/Do=1.3S/Do=1.2


311

Figure 11. Test setup for both the optimized castellated beam with sinusoidal opening

Figure 12. Load vs. deflection curve for optimized castellated beam with sinusoidal opening

From above graphs, it is observed that optimized beam with sinusoidal opening takes the

load of 34kN causing the deflection of 1.42 mm.

7. COMPARATIVE STUDY OF CASTELLATED BEAM WITH CIRCULAR

AND DIAMOND SHAPED OPENING

After finding out optimized dimension for sinusoidal shaped castellated beam it is very

important to choose the optimized shape for openings. In order to find out optimized shape

comparison of results of hexagonal and sinusoidal shaped openings need to be done. This

comparison of results of castellated beam with hexagonal and sinusoidal shaped openings is

given in Table 3. Below;

Table 3: Failure load and deflection of castellated beam with type of openings

Sr.

No.

Type of castellated

beam (shape wise)

Failure load by von-

misses criteria (kN)

Deflection by

FEA (mm)

Deflection by

experiment (mm)

1 Hexagonal opening [13] 30 1.3 1.36

2 Sinusoidal opening 34 1.594 1.42

34 ؛1.42

0

10

20

30

40

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Load

(k

N)

Deflection (mm)

Optimized Castellated Beam with Sinusoidal Opening


312

From Table 3, it be concluded that castellated beams with sinusoidal shaped openings is

more reliable from strength requirement point of view. In the other words, the castellated

beam with sinusoidal shaped opening with D/Do ratio of 1.41 and S/Do ratio of 1.4 gives

more satisfying strength results than the other shapes and sizes of castellated beam.

8. CONCLUSIONS

Following conclusions can be drawn from the study:

1. Castellated beam with sinusoidal openings of size of 0.55 times its overall depth with

S/Do ratio of 1.4 and D/Do ratio of 1.41 of takes maximum load of 34 kN.

2. As in case of sinusoidal shaped openings more shear transfer area is available,

therefore castellated beam with sinusoidal openings proves to be better than the other shaped

openings in respect of taking loads.

3. Sinusoidal openings gives the curved edges instead of corners. Therefore, the stress

distribution near the corner portion of opening is uniform resulting in less stress

concentration at opening.

4. Results of Abaqus software (FEA) are in good agreement with the results of

experimentation and also with method of analysis given by Eurocode.

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1. Boyer JP. Castellated beam – new developments, Proceedings of AISC National

Engineering Conference 1964; 3: pp. 106-108.

2. British Standard Institution. Design of steel structures, Part 1.1: General rules and rules for

buildings, Eurocode 3, ENV 1993-1-1, 1992.

3. British Standard Institution. Structural use of steelwork in building, Part 1: code of practice

for design rolled and welded sections, BS 5950-1, 2000.

4. Durif S, Bouchaïr A. Behaviour of cellular beams with sinusoidal openings, Steel Struct

Bridges, Proced Eng 2012; 40: 108-13.

5. Erdal F, Saka MP. Ultimate load carrying capacity of optimally designed steel cellular

beams, J Construct Steel Res 2013; 80: 355-68.

6. Kaveh A, Shokohi F. Cost optimization of castellated beams using charged system search

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Edition, CRC Press, New York, 2013.

8. Lawson RM, Hicks SJ. Design of Composite Beams with Large Openings, SCI P355, SCI

Publication, Berkshire UK, 2011.

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10. Raftoyiannis IG, Ioannidis GI. Deflection of castellated I beam under transverse loading,

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